New closed forms for a class of digamma series and integrals

The pursuit of closed forms for infinite series has long been a focal point of research. In this paper, we contribute to this endeavor by presenting closed forms for the class of digamma series: $$\sum_{k=1}^\infty \frac{\psi\left(\frac{2k+2n+5}{4}\right) - \psi\left(\frac{2k+2n+3}{4}\right)}{(2k + \alpha)^2},$$ $$\sum_{k=1}^\infty (-1)^k \frac{\psi\left(\frac{2k+2n+5}{4}\right) - \psi\left(\frac{2k+2n+3}{4}\right)}{(2k + \alpha)^2},$$ for all non-negative integers $\alpha$ and $n$. In addition to providing closed forms for these series, we unveil new identities for various generalized digamma series in the elegant form $a_0 + a_1 \pi + a_2 \pi^2 + a_3 \pi^3$, where $a_0, \ldots, a_3$ are real-valued constants determined by our formulas. Furthermore, we present ten definite integrals over the interval $(0, 1)$ that have not been previously studied in the literature and appear to be nearly impossible to evaluate. Combining these series and integrals can lead to the discovery of even more new results. Our findings contribute to the study of closed forms for infinite series and integrals, offering novel results and potential avenues for further exploration.Comment: 33 page

Contextual Search in the Presence of Irrational Agents

We study contextual search, a generalization of binary search in higher dimensions, which captures settings such as feature-based dynamic pricing. Standard game-theoretic formulations of this problem assume that agents act in accordance with a specific behavioral model. In practice, however, some agents may not prescribe to the dominant behavioral model or may act in ways that are seemingly arbitrarily irrational. Existing algorithms heavily depend on the behavioral model being (approximately) accurate for all agents and have poor performance in the presence of even a few such arbitrarily irrational agents. We initiate the study of contextual search when some of the agents can behave in ways inconsistent with the underlying behavioral model. In particular, we provide two algorithms, one built on robustifying multidimensional binary search methods and one on translating the setting to a proxy setting appropriate for gradient descent. Our techniques draw inspiration from learning theory, game theory, high-dimensional geometry, and convex analysis.Comment: Compared to the first version titled "Corrupted Multidimensional Binary Search: Learning in the Presence of Irrational Agents", this version provides a broader scope of behavioral models of irrationality, specifies how the results apply to different loss functions, and discusses the power and limitations of additional algorithmic approache

Essays on Dynamic Mechanism Design

We consider the allocation of one or several units of a good in a dynamic environment. The time horizon is finite and in each period, a random number of potential buyers arrives. In Chapter 1, we study revenue maximization in an environment where buyers are privately informed about their valuations and their deadlines. Depending on the type distribution, the incentive compatibility constraint for the deadline may or may not be binding in the optimal mechanism. We identify a static and a dynamic pricing effect that drive incentive compatibility and violations thereof. Both effects are related to distinct properties of the type distribution and sufficient conditions are given under which each effect leads to a binding or slack incentive constraint for the deadline. An optimal mechanism for the binding case is derived for the special case of one object, two periods and two buyers. It can be implemented by a fixed price in period one and an asymmetric auction in period two. In order to prevent buyer one from buying in the first period when his deadline is two, the seller sets a reserve price that is lower than in the classic (Myerson, 1981) optimal auction and gives him a (non-linear) bonus. The bonus leads to robust bunching at the top of the type-space. Chapter 2 contains a characterization of asymmetric reduced form auctions. In chapter 3, we consider a more general dynamic environment in which buyers' valuations may depend on the time of allocation in an arbitrary way. We show that the static Vickrey auction can be generalized to the dynamic framework. This yields a simple payment rule for the implementation of the efficient allocation rule of a single object. To define the dynamic Vickrey auction, we show that the multi-dimensional type-space can be reduced to essentially one dimension. This allows to define the winner's payment as the lowest valuation in the reduced type-space, that suffices to win. Finally we define an ascending clock-auction with an equilibrium-outcome that coincides with the outcome of the dynamic Vickrey auction

Proceedings of the second "international Traveling Workshop on Interactions between Sparse models and Technology" (iTWIST'14)

The implicit objective of the biennial "international - Traveling Workshop on Interactions between Sparse models and Technology" (iTWIST) is to foster collaboration between international scientific teams by disseminating ideas through both specific oral/poster presentations and free discussions. For its second edition, the iTWIST workshop took place in the medieval and picturesque town of Namur in Belgium, from Wednesday August 27th till Friday August 29th, 2014. The workshop was conveniently located in "The Arsenal" building within walking distance of both hotels and town center. iTWIST'14 has gathered about 70 international participants and has featured 9 invited talks, 10 oral presentations, and 14 posters on the following themes, all related to the theory, application and generalization of the "sparsity paradigm": Sparsity-driven data sensing and processing; Union of low dimensional subspaces; Beyond linear and convex inverse problem; Matrix/manifold/graph sensing/processing; Blind inverse problems and dictionary learning; Sparsity and computational neuroscience; Information theory, geometry and randomness; Complexity/accuracy tradeoffs in numerical methods; Sparsity? What's next?; Sparse machine learning and inference.Comment: 69 pages, 24 extended abstracts, iTWIST'14 website: http://sites.google.com/site/itwist1

“TEACHING REAL NUMBERS IN THE HIGH SCHOOL: AN ONTO-SEMIOTIC APPROACH TO THE INVESTIGATION AND EVALUATION OF THE TEACHERS' DECLARED CHOICES”

The thesis addresses the topics of investigating teachers' declared choices of practices concerning real numbers and the continuum in the high school in Italy, evaluating their didactical suitability and the impact of a deep reflexion about some historical and didactical issues on the teachers' decision-making process. Our research hypothesis was that teachers' choices of teaching sequences concerning real numbers, with particular attention to the representations of real numbers, could be very relevant in order to interpret some of the well-known students' difficulties. After a pilot study in form of a teaching experiment and a literature review concerning students' and teachers' difficulties with real numbers and the continuum, we observed that some causes of potential difficulties could be situated indeed in the very beginning of the teaching-learning process, even before entering the classrooms: the phase in which a teacher choose the practices and objects by means of whom introducing and work with real numbers and the continuum. In particular the choice of the objects involved in the practice seemed to be relevant, since every object emerge from previous practices and its meaning is identified by the practices in which it emerged. Thus we got interested in investigating the personal factors that affect the process of selection of practices: personal meaning, goals and orientations, as it was stressed by Schoenfeold in his goal-oriented decision-making approach to the analysis of teachers choices. Furthermore we decided to explore the teachers' choices of sequences of practices and of representation of the mathematical objects and then to evaluate their suitability in relation to the literature review concerning students' difficulties with real numbers and to the complexity of the mathematical object as it emerge from an historical analysis. After having analysed the theoretical frameworks in mathematics education that could permit us to carry out our research, we decided to use the OSA, (onto-semiotioc approach) elaborated by Godino, Batanero & Font, described in their paper in 2007, and some evolutions like the CDM (conoscimiento didactico matematico) model proposed by Godino in 2009. We evaluated also other frameworks, in particular the ATD (Chevallard, 2014), but we found the OSA better for the analysis we would like to carry out. In particular the operationalization of the methodologies of analysis of the teachers' personal meaning of mathematical objects and the construct of didactical suitability were more effective for our porpouses. Our main results are the following: mny teachers' personal meanings of real numbers are far from the epistemic one; many of the teachers who studied real numbers at a formal level at school and at the University and perceived them as difficult and unuseful try to avoid to deepen the issues concerning real numbers with their stundent, thinking they would not understand; in general the experiences as students affect the teachers' choices; the teachers usually refer to real numbers also when the meaning is partial and doesn't coincide with one of the most general epistemic meanings of real numbers; very few teachers are aware of the complexity of the real numbers and are as aware of it to be able to control the relations between their many facets; also the teachers with a PhD in Mathematics operate choices that we can evaluate as unsuitable standing on the literature review and our framework; the teacher consider very hard to work with discrete and dense sets and prefer the intuitive approach to continuous sets rather then deepen the relation between dense and continuous sets, different degrees of infinity and so on; some teachers reasoning during the interviews changed their mind, getting aware of the complexity and admitting that simplifying too much can constitute a further cause of difficulty; the teachers refer to the students difficulties to justify their choice of simplifying, but when they face some crucial issues, often they admit to consider them unuseful or too difficult; nevertheless no teachers declare that would renounce to introduce the field of real numbers, at least intuitively; the most of the teachers declare that nothing more is introduced about real numbers in the last years and that the partial meanings introduced in the first years are used to face the Calculus problems (intuItive approach to the Calculus); all the teachers consider necessary to introduce R or adequate subsets of R as domains of the functions expressed analytically because of their continuous graphic.The thesis addresses the topics of investigating teachers' declared choices of practices concerning real numbers and the continuum in the high school in Italy, evaluating their didactical suitability and the impact of a deep reflexion about some historical and didactical issues on the teachers' decision-making process. Our research hypothesis was that teachers' choices of teaching sequences concerning real numbers, with particular attention to the representations of real numbers, could be very relevant in order to interpret some of the well-known students' difficulties. After a pilot study in form of a teaching experiment and a literature review concerning students' and teachers' difficulties with real numbers and the continuum, we observed that some causes of potential difficulties could be situated indeed in the very beginning of the teaching-learning process, even before entering the classrooms: the phase in which a teacher choose the practices and objects by means of whom introducing and work with real numbers and the continuum. In particular the choice of the objects involved in the practice seemed to be relevant, since every object emerge from previous practices and its meaning is identified by the practices in which it emerged. Thus we got interested in investigating the personal factors that affect the process of selection of practices: personal meaning, goals and orientations, as it was stressed by Schoenfeold in his goal-oriented decision-making approach to the analysis of teachers choices. Furthermore we decided to explore the teachers' choices of sequences of practices and of representation of the mathematical objects and then to evaluate their suitability in relation to the literature review concerning students' difficulties with real numbers and to the complexity of the mathematical object as it emerge from an historical analysis. After having analysed the theoretical frameworks in mathematics education that could permit us to carry out our research, we decided to use the OSA, (onto-semiotioc approach) elaborated by Godino, Batanero & Font, described in their paper in 2007, and some evolutions like the CDM (conoscimiento didactico matematico) model proposed by Godino in 2009. We evaluated also other frameworks, in particular the ATD (Chevallard, 2014), but we found the OSA better for the analysis we would like to carry out. In particular the operationalization of the methodologies of analysis of the teachers' personal meaning of mathematical objects and the construct of didactical suitability were more effective for our porpouses. Our main results are the following: mny teachers' personal meanings of real numbers are far from the epistemic one; many of the teachers who studied real numbers at a formal level at school and at the University and perceived them as difficult and unuseful try to avoid to deepen the issues concerning real numbers with their stundent, thinking they would not understand; in general the experiences as students affect the teachers' choices; the teachers usually refer to real numbers also when the meaning is partial and doesn't coincide with one of the most general epistemic meanings of real numbers; very few teachers are aware of the complexity of the real numbers and are as aware of it to be able to control the relations between their many facets; also the teachers with a PhD in Mathematics operate choices that we can evaluate as unsuitable standing on the literature review and our framework; the teacher consider very hard to work with discrete and dense sets and prefer the intuitive approach to continuous sets rather then deepen the relation between dense and continuous sets, different degrees of infinity and so on; some teachers reasoning during the interviews changed their mind, getting aware of the complexity and admitting that simplifying too much can constitute a further cause of difficulty; the teachers refer to the students difficulties to justify their choice of simplifying, but when they face some crucial issues, often they admit to consider them unuseful or too difficult; nevertheless no teachers declare that would renounce to introduce the field of real numbers, at least intuitively; the most of the teachers declare that nothing more is introduced about real numbers in the last years and that the partial meanings introduced in the first years are used to face the Calculus problems (intutive approach to the Calculus); all the teachers consider necessary to introduce R or adequate subsets of R as domains of the functions expressed analytically because of their continuous graphic

Irony and ambiguity in Beethoven's string quartets

This thesis explores the view that many of the difficulties and apparent eccentricities of Beethoven's Late Quartets (particularly Op. 130, 132, 133 and 135) may be understood in terms of irony, in the sense that it appears in the philosophical and aesthetic writings of the early German Romantics. A chain of influence is demonstrated between Beethoven and Friedrich Schlegel's philosophy of Romantic irony, through significant inter -personal relationships as well as through Beethoven's exposure to Schlegel's written works. This connection provides a firm hermeneutic basis for considering the composer's work in terms of irony.The A minor Quartet Op. 132 is given as an example of Beethoven's Romantic irony, and considered in terms of the constitutive elements of Schlegel's Romantic irony - Paradox, Parabasis and Self -consciousness. However, this thesis also demonstrates that the irony within the Late Quartets goes beyond the confines of Romantic irony. The paradoxical structures of the Cavatina and Grosse Fuge are considered as examples of "general" or "existential" irony -a form closely related to Schlegelian irony. Moreover, the replacement finale of the Op. 130 quartet is shown to constitute a striking instance of satire: a bitter ironic comment upon the musical conservatism of Beethoven's critics.This thesis therefore explores the philosophical background and the nature of irony itself, relating all of its forms to one underlying structure and to one fundamental process. This process - "objectification" - is derived from the work of Mikhail Bakhtin, and forms the theoretical basis for the structural approach of the analyses of irony within the thesis. The thesis also considers the relationship between irony and related phenomena such as wit and humour. It suggests that the differences between these concepts correspond to those between Beethoven's Romantic irony and the wit and humour of his predecessors.Finally, the relationship between irony and ambiguity is also considered. Ambiguity is frequently elided with irony within theoretical writing on irony; indeed the terms "irony" and "ambiguity" are often used synonymously. Since ambiguity is a significant element of the harmonic and formal practices within the Quartets this elision is important: if ambiguity and irony are elided then each instance of ambiguity may be considered ironic - a reductio ad absurdum. This work distinguishes ambiguity and irony as separate phenomena, approaching this division through the semiotic concepts of "immanence" and "manifestation ". I argue that ambiguity occurs as a particular effect of the immanent level of discourse, whilst irony occurs entirely within the manifest level. In addition to this difference in function, different structures are demonstrated for these phenomena. This distinction is applied to the third movement of the Op. 130 Quartet, which is considered as a confrontation of Classical aesthetics with the equivocal and ambiguou

Moment-Based Estimation of Macroscopic Dynamic Models in Macroeconomics and Finance

The thesis is divided into two parts. Part I (macroeconomic models) is devoted to empirically examine the role of backward-looking behavior in a standard New-Keynesian model and its behavioral variant. This part includes three chapters. In chapter 2, the structural parameters of the New-Keynesian model (NKM) are estimated from a historical data set of the US economy. The moment-matching method is used to discuss the importance of backward-looking behavior in the New-Keynesian Phillips Curve, and its empirical results are contrasted to the Bayesian estimation. Then the model with purely forward-looking expectations and its hybrid variant are compared using a formal test. Chapter 3 discusses the persistent dynamics of inflation and output in the NKM, and analyzes statistical properties of the estimation methods and the choice of moment conditions. The empirical performance of model selection methods is examined using information criteria along the lines of the maximum likelihood estimation. Chapter 4 demonstrates that market euphoria and depression in the economy can be explained by switching between heterogeneous investors along the lines of the discrete choice theory. This chapter investigates a bounded rationality model with historical Euro Area data and discuss the effects of the investors' over(or under)reaction on market structure. Part II (financial market models) contains two chapters concerning the relevance of behavioral heterogeneity in financial markets. Chapter 5 discusses the effect of heterogeneous trading rules on the return volatility in the asset pricing model. The adaptive belief system is estimated using the simulated method of moments estimator. Especially, two types of the noise term in the model dynamics are investigated by means of simulations (i.e.\ additive and multiplicative). In a structural stochastic volatility model, the two trading mechanisms (i.e.\ wealth and herding) are compared according to a formal test. Finally, chapter 6 examines the social interaction effects of a market microsimulation model on various historical FX data. In particular, simulation based inference is used to examine the validity of the group behavior when the analytical expression for moment conditions is fairly complicated